Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis)

Book Description

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.

After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.

As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.

Also available, the first two volumes in the Princeton Lectures in Analysis:

Customer Reviews:

great book.......2006-10-19

i found the first three chapters of this book very clear and well written. i'd strongly recommend it for someone looking to learn about analysis on the real line.

Good book for reading and as a graduate student.......2006-07-19

Easy to read. My university is using this book to get the graduate students ready for the real analysis qualifying exam. So go ahead and buy this book if you're planning to work on a PhD in mathematics. If you're not planning to work on a PhD in math, this is still a good book to read if you enjoy studying about the real line.

The book begins with measure theory, integration and differentiation. These are included in Chapters 1 to 3. Then in Chapters 4 and 5, we look into Hilbert spaces. This is similar to studying finite-dimensional inner-product spaces, but here, Hilbert space is infinite-dimensional. However, the analysis is very similar. If you know some linear algebra, it should feel like as if you have already read these two chapters.

Finally in Chapters 6 and 7, we see abstract measure theory, including Hausdorff measure, and we study fractals and self-similar sets. And this concludes the book.

Also recommend Walter Rudin's Real Analysis.

Suffers from all the flaws of a 1st edition.......2005-12-18

This book has a lot of problems. Several sections are poorly written/edited. Several important named theorems are not clearly labeled. Also some of the proofs contain typos or errors. The chapter on differentiation is particularly lacking. The chapter is poorly organized and presented. There is also a glaring TeX error in the chapter.

At Princeton this book is used as part of an undergraduate course, and it shows. This is not the ideal book for a graduate level course in real analysis(though I think it would be very well suited for advanced undergrads). Too much time is spent on Lebesgue measure and integration in the first 2 chapters, and abstract measure theory is not intoduced until chapter 6. Also the Monotone Class theorem is lacking from the chapter on abstract measure theory. Also, the book only touches on functional analysis in the two chapters on Hilbert spaces (where they assume all Hilbert spaces are separable).

On the other hand, the presentations of Lebesgue measure/integration and Hilbert spaces in the book are pretty good. The exercises and problems in teh book (when stated properly) are very good and instructive. Overall this book has a lot of potential to be very good, but seems to be suffering from a lack of revision. Hopefully these issues will be fixed in later editions.

Excellent sourse for graduate analysis.......2005-07-03

This book is the best book on real analysis I have ever studied. It does a wonderful job in bridging undergraduate level with graduate level analysis. I have not seen any book that makes measure and Lebesgue theory so easy to understand.

The books begins by defining what a "measure" is all about. And the description is so intuitive and geometrical that you would wonder why you weren't taught it this way before. The book then goes into Lebesgue theory and all of it suddenly becomes so easy.

The book has plenty of wonderful examples and a good set of over 30 problems per chapter.

Elias Stein (one of the authors) is a very renowned mathematician, and one need not worry about the accuracy of the proofs in the book--they are "bullet-proof", and at the same time succinct.

If you are struggling with W. Rudin's book on Analysis, this book is a MUST for you.

Measure Theory and Integration (Pure and Applied Mathematics)

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Jon's Review

Measure Theory and Integration (Pure and Applied Mathematics)
M. M. Rao
Manufacturer: Marcel Dekker
ProductGroup: Book
Binding: Hardcover

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A Modern Theory of Integration (Graduate Studies in Mathematics)

ASIN: 0824754018

Book Description

Significantly revised and expanded, this authoritative reference/text comprehensively describes concepts in measure theory, classical integration, and generalized Riemann integration of both scalar and vector types-providing a complete and detailed review of every aspect of measure and integration theory using valuable examples, exercises, and applications. With more than 170 references for further investigation of the subject, this Second Edition · provides more than 60 pages of new information, as well as a new chapter on nonabsolute integrals · contains extended discussions on the four basic results of Banach spaces · presents an in-depth analysis of the classical integrations with many applications, including integration of nonmeasurable functions, Lebesgue spaces, and their properties · details the basic properties and extensions of the Lebesgue-Carathéodory measure theory, as well as the structure and convergence of real measurable functions · covers the Stone isomorphism theorem, the lifting theorem, the Daniell method of integration, and capacity theory Measure Theory and Integration, Second Edition is a valuable reference for all pure and applied mathematicians, statisticians, and mathematical analysts, and an outstanding text for all graduate students in these disciplines.

Customer Reviews:

Jon's Review.......2002-04-18

Simply put, M.M. Rao's "Measure Theory and Integration" is an awesome book. It is truly the "Encyclopedia Britannica" of Real Analysis textbooks. This math textbook/reference book contains the most general, yet practical, theorems on the subject known to mankind. I cannot recommend it highly enough.

Average customer rating:

Really Great for Certain Topics
Maybe good as a supplement, or a first time looking at the material
Classic text on measure & integration theory
Not very instructive to newcomers
Not perfect, but better than the rest

Real Analysis (3rd Edition)
Halsey Royden
Manufacturer: Prentice Hall
ProductGroup: Book
Binding: Hardcover

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ASIN: 0024041513

Customer Reviews:

Really Great for Certain Topics.......2007-08-19

Great for the bookshelf but really pretty hard - you need a good proof course and a year suffering through baby Rudin - and believe me, you will suffer - but will be a better person (and mathematician) for it.

Maybe good as a supplement, or a first time looking at the material.......2007-02-09

There are three books that are usually used for a first graduate course in analysis, including measure theory, namely Rudin's Real and Complex Analysis, Folland's Real Analysis, and Royden's book. Of the three, I would say Royden's book is the easiest, both in terms of the exposition, material, and exercises. Of the three, Royden is the only one to fully develop the Lebesgue measure and the associated integral before developing a more general theory of measure and integration. Furthermore, he does not develop Hilbert and Banach space theory, the very basics of functional analysis, to anywhere near the extent that Folland and Rudin do.

There is some debate as to whether it is better to start with the Lebesgue integral, and then talk about abstract integration, or the other way around. Personally, I found the development of the Lebesgue integral a bit tedious; the whole thing works a bit better when you first talk about abstract integration, which really isn't a terribly difficult concept, prove the basic integration theorems, then show how to construct an outer measure, and suddenly, the Lebesgue measure and integral just falls into place. I'm not sure anything is lost in the process.

The biggest shortcoming in this book would have to be the exercises: for the most part, they are not very difficult, particularly when you compare them to say, Rudin's text. For the most part, the exercises are fairly trivial, and if they are difficult, or require a bit of creativity, Royden often gives you lots and lots of hand-holding, sometimes even in the form of sketching out the proof for you. In spite of the relatively low difficulty level, most of the exercises are fairly instructive, in so far as they highlight, elucidate, and expand upon the material.

For the most part, this book is not bad. It makes a good supplement to a book like Rudin or Folland, as it is less abstract, and does a better job motivating the material. The exercises here can work well if you want some extra practice that won't take up too much time. If you're a student of econ, or physics, or you just feel like learning graduate-level real analysis, then this book is probably adequate (although I should qualify that statement by saying that I know nothing of econ and little of physics). But if you are a serious student of mathematics, particularly the pure variety, this is really not the book you should be using. It is just too easy.

Classic text on measure & integration theory.......2006-08-23

Many people criticize this book as unclear and unnecessarily abstract, but I think these comments are more appropriately directed at the subject than at this book and its particular presentation. I find this classic to be one of the best books on measure theory and Lebesgue integration, a difficult and very abstract topic. Royden provides strong motivatation for the material, and he helps the reader to develop good intuition. I find the proofs and equations exceptionally easy to follow; they are concise but they do not omit as many details as some authors (i.e. Rudin). Royden makes excellent use of notation, choosing to use it when it clarifies and no more--leaving explanations in words when they are clearer. The index and table of notation are excellent and contribute to this book's usefulness as a reference.

The construction of Lebesgue measure and development of Lebesgue integration is very clear. Exercises are integrated into the text and are rather straightforward and not particularly difficult. It is necessary to work the problems, however, to get a full understanding of the material. There are not many exercises but they often contain crucial concepts and results.

This book contains a lot of background material that most readers will either know already or find in other books, but often the material is presented with an eye towards measure and integration theory. The first two chapters are concise review of set theory and the structure of the real line, but they emphasize different sorts of points from what one would encounter in a basic advanced calculus book. Similarly, the material on abstract spaces leads naturally into the abstract development of measure and integration theory.

This book would be an excellent textbook for a course, and I think it would be suitable for self-study as well. Reading and understanding this book, and working most of the problems is not an unreachable goal as it is with many books at this level. This book does require a strong background, however. Due to the difficult nature of the material I think it would be unwise to try to learn this stuff without a strong background in analysis or advanced calculus. A student finding all this book too difficult, or wanting a slower approach, might want to examine the book "An Introduction to Measure and Integration" by Inder K. Rana, but be warned: read my review of that book before getting it.

Not very instructive to newcomers.......2006-05-11

I'm currently earning an MS in Mathematics, and am currently completing the last few weeks of my Real Analysis course using this book as its text. On the whole, I don't like this book.

I'm sure to people whom analysis comes naturally this is a fine book, and they can learn a lot from it. Also, I believe that a good professor could deliver Royden's text so that any student would have a good time learning it.

However I'm not a natural analyst, and in the absence of a good teacher I'm forced to rely on the text alone. I have had a terrible time, specifically:
-Incessant omissions in homework problems leave me wondering under what conditions I'm proving the theorem.
-the book references some fairly obscure ideas without explanation (which analyists may be familiar with, but I have no idea about, e.g. "a standard diagonal argument")
-almost no time is spent discussing common methods for proving certain species of analytical proofs
-some of the notation is outdated (according to at least one professor) and could be cleaned up
-I think one too many important proofs were "left to the reader"

On the upside:
-On the whole, the proofs are interesting and diverse
-the problems I did understand taught me much
-Royden's repetition is a good learning device and makes it easier to find some information

Not perfect, but better than the rest.......2005-08-23

I'm a PhD student in mathematics at Georgia Tech. I used this when I first took graduate real analysis at North Dakota State, and then used Wheeden and Zygmund's Measure and Integral here at Georgia Tech as well as Folland's Real Analysis when studying for comps. Time and time again, I found myself going back to Royden for his well-written expositions that left enough out to keep you paying attention but wasn't so sparse that you couldn't figure out what was going on. Some here have complained about it doing everything twice. This can be a problem in some cases, such as common texts for a first course in real analysis where topological ideas are covered for Euclidean space first and then again for general metric spaces, but with measure theory, this is the right approach. I saw it first hand last fall, as my colleagues in another section treated Lebesgue measure on the real line as a special case and did things in generality, while my section dealt with R^n first and then moved on to general measures. In the end, I'm quite sure the section that I was in had a firmer grasp on the material.

Royden's classic work has withstood the test of time, and deserves to remain a standard text for years to come.

Average customer rating:

An exceptionally good book
The book on probability
Some nice examples, poorly organized
a very good text book
Standard text but ...

Probability and Measure, 3rd Edition
Patrick Billingsley
Manufacturer: Wiley-Interscience
ProductGroup: Book
Binding: Hardcover

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A Course in Probability Theory Revised
Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics)
First Look at Rigorous Probability Theory
A Probability Path
Testing Statistical Hypotheses (Springer Texts in Statistics)

ASIN: 0471007102

Book Description

PROBABILITY AND MEASURE

Third Edition

Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory.

Like the previous editions, this new edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.

Customer Reviews:

An exceptionally good book.......2006-12-19

I've read portions of almost every measure theoretic probability theory book published. And I've come back to Billingsley. This is a hard book to read through and through if you are a novice; this is not Billingsley's fault - it is just that the subject is hard on first acquaintance.
Billinglsey develops everything from first principles, so if you have the intellectual gumption you ought to be able to read the main text with a knowledge of plain college algebra and a little epsilon-delta practice of the sort that comes from an undergraduate real analysis course. The small print asides are fascinating but they are often addressed to a card carrying mathematician. The November 2003 reviewer who complained that Billingsley uses expectation before defining the integral fails to notice - or at any rate, to point out - that he defines only the expectation of simple random variables in the first chapter, so what is involved is just a sum, not an integral. I could sing my praises on and on. But here is the kernel of this review in a line: this is one of the best books ever written on measure theoretic probability. Full stop.

The book on probability.......2006-01-26

This book is not for everybody. It is for the professional mathematician (or physicist, or alike). All concepts are very well explained, and Billigsley does go down to the core of everything. It is, as far as I'm concerned, among the best books in math ever written, with favorites such as Feynman's lectures and Herstein's algebra manual. If you are a mathematician and want to have the top reference in probability, this is it.

Some nice examples, poorly organized.......2003-11-27

This is probably a very nice text book if you already know probability. There are undeniably some insightful examples. However, it is often hard to follow the sequence of topics in the book.

It is at least amusing that the integral is only developed a couple of chapters after expectation has been in use...

a very good text book.......2003-09-12

This book gives abundant examples and statements that help to deepen your understanding. It does not require much statistic background to follow the book, although sometimes the reasoning is not that obvious to me. But maybe because I am not a math student. I also feel that topics are a bit scattered in the book.

Standard text but ..........2003-06-19

the main problem with Billingsley's book lies in its organization of topics and results.
Yes it has all the standard results that need to be covered in a first (rigorous) course on probability theory and the proofs and exercises are good (thats why the three stars) but it is incredibly hard to study them from this book because of poor organisation which makes for lack of continuity (thats why no more than three).
Stick to Chung (and move to something more specialized thereafter). Unfortunately, Parthasarathy's 1977 Macmillan book is now out of print and only available in libraries ... I find that to be the best book at this level.

Taking the Measure of Work; A Guide to Validated Scales for Organizational Research and Diagnosis

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Taking the Measure of Work; A Guide to Validated Scales for Organizational Research and Diagnosis
Dail L. Fields
Manufacturer: Sage Publications
ProductGroup: Book
Binding: Hardcover

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Great Writers on Organizations

ASIN: 0761924256

Book Description

"It is well, well done -- I will indeed recommend it . . . this type of work has been long needed in our field."
--Robert J. Vandenberg, University of Georgia

Organizational researchers and managers have never had a single easy-to-use resource for validated measures, often relying on a selection of journal articles or improvised solutions to meet immediate needs. Taking the Measure of Work: A Guide to Validated Scales for Organizational Research and Diagnosis provides researchers, consultants, managers, and organizational development specialists validated and reliable ways to measure how employees view their work and their organization.

Whether preparing questionnaires or interviews for an employee survey, organizational assessment, dissertation or research program, this book guides users to a summary level understanding of each topic area, the measurement issues in the area, and a selection of measures to choose from. The measures cover the areas of:

Job Satisfaction
Organizational Commitment
Job Characteristics
Job Stress
Job Roles
Organizational Justice
Work-Family Conflict
Person-Organization Fit
Work Behaviors
Work Values

About the Author

Dail L. Fields (Ph.D., Georgia Tech, 1994) is Associate Professor at the Regent University School of Business. His research interests include measurement of employee perspectives on work, cross-cultural management, human resource management strategies, and leadership and values in organizations. He is a member of the Academy of Management and the Academy of International Business. Prior to beginning an academic career in 1994, he was a management executive with MCI Communications Corp. and a management consultant with Touche Ross & Co.

Measure Theory and Fine Properties of Functions (Studies in Advance Mathematics)

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Inspiring and Useful

Measure Theory and Fine Properties of Functions (Studies in Advance Mathematics)
Lawrence C. Evans
Manufacturer: CRC Press
ProductGroup: Book
Binding: Hardcover

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Geometric Measure Theory: A Beginner's Guide

ASIN: 0849371570

Book Description

This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the fine properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. The text provides complete proofs of many key results omitted from other books, including Besicovitch's Covering Theorem, Rademacher's Theorem (on the differentiability a.e. of Lipschitz functions), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions). Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in applied mathematics.

Customer Reviews:

Inspiring and Useful.......2000-06-13

I was turned on to this book by a friend of mine who is an expert in geometric measure theory. He recommended the book as a very nice exposition of some of the material found in Federer's "Geometric Measure Theory" as well as other material. I found the book to be beautifully designed to help the reader learn its contents. There was enough between the lines so that one needed to WORK through the book, but in contrast to parts of Federer's book, enough detail so that reasonably fast progress could be made. Unfortunately, I was interupted in my race through the book and so I have yet to work through the latter part of the book. But given the large part I did cover and my experience doing that, I am certain to finish the monograph, most likely when I start using functions of bounded variation with any frequency.

There are no explicit exercises. But as already alluded to above, there are implicit exercises that are encountered in working through the book. I found that the lack of separate exercises is actually not bad at all since the implicit exercises encountered are automatically motivated by their necessity for the understanding of the text - and are therefore relevant!

A prerequisite for the book is a course in analysis that includes measure theory and integration as well as an exposure to elementary functional analysis. The functional analysis is not actually necessary, but the added maturity that such an exposure would impart would be useful.

Very briefly, the contents via the 6 chapter titles are 1) General Measure Theory, 2) Hausdorff Measure, 3) Area and Coarea Formulas, 4) Sobolev Functions, 5) BV Functions and Sets of Finite Perimeter, and 6) Differentiability and Approximation by C^1 Functions.

I found the contents very interesting ... quoting the authors "... we packed into these notes all sorts of interesting topics that working mathematical analysts need to know, but are mostly not taught." And indeed this was the case in my experience ... both the "interesting" part and the "not taught" part.

Book Description

Public and private entities are rapidly developing new methods for measuring and reviewing psychiatric care—methods that may not always reflect good research into the perspective of clinicians. Mental health providers and their patients are being held to new "criteria" in determinations of access to services. A wide range of clinical and policy issues are affected by the selection and application of psychiatric measures: eligibility determinations, outcomes assessment, pricing, and risk adjustment, as well as activities related to quality assurance and utilization review. The Handbook of Psychiatric Measures responds to these challenges.

The handbook provides clinicians working in mental health or primary care settings with a compendium of the available rating scales, tests, and measures that may be useful for caring for patients with mental illnesses. In addition, it provides guidance to clinicians, policy makers, and planners on how to better understand and properly use clinical measures to assess performance of individual providers or groups of providers in health care delivery systems.

In selecting, applying, and interpreting a measure, clinicians must be knowledgeable about the nuances of measurement, including not only the psychometric properties of an instrument (e.g., its reliability or validity), but, even more importantly, the factors that affect the clinical utility of the measure. This handbook supplies that knowledge, with detailed information about components, reliability, validity, and clinical utility, including strengths and weaknesses, for each measure included.

Accompanying this handbook is a CD-ROM that includes complete copies of 108 measures discussed in the handbook—over 900 pages of measures. The CD-ROM also includes the complete, unabridged text of the Handbook of Psychiatric Measures, in fully searchable form. The electronic version is replete with links and cross-references, including links from the text of the handbook to the actual measures being discussed.

Customer Reviews:

Value in the Handbook of Psychiatric Measures.......2006-03-16

I purchased this book because of my studies in family therapy in particular family therapy with young clients who have a developing mental illness.
This book will assist in clarifying issues and lead to a speedy resolution of their problems, and assist families in helping their family member in the recovery process

A Disappointment.......2003-11-14

The book is good..... The CD that is included containing the actual tests, inventories and measures on it is nearly unacceptable as far as quality. The scanning job was poor and there is nothing that can be done to improve the quality or change the size. if I had seen it prior to purchase, I wouldn't have bought it. Additionally, when I contacted the publisher, the customer servifce representative was not even remotely interested in my concern or feedback.

Excellent resource for health scales.......2003-05-22

I was primarily interested in social health scales, and found this book to be an invaluable listing of many of the scales available, including reliability and validity information, and actual scale text on e-file with the attached CD. Not for psychiatrists only, for any health researcher interested in using tested scales.

Average customer rating:

All background needed for Ito calculus is here
All backgound needed for Ito calculus is here!

Probability Essentials
Jean Jacod
Manufacturer: Springer-Verlag New York, Inc
ProductGroup: Book
Binding: Paperback

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ASIN: 3540438718

Book Description

This introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. The second edition contains some additions to the text and to the references and some parts are completely rewritten.

Customer Reviews:

All background needed for Ito calculus is here.......2000-10-04

This is an excellent and timely textbook on probability and martingale theory. There is an increasing need of thorough but concise treatise of probability theory for researchers and graduate students in Engineering, Economics, Statistics and Mathematical Biology. Very few textbook fill this need. Jacod and Protter succeeded in bringing together essential concepts and theorems in probability/martingale theory in a clear and lucid style and the book is completely self-contained: all necessary machinery from measure theory are explained and proved while providing a flavor of probabilistic way of thinking. Unlike Williams' "Probability with Martingales", all mathematical details are covered in the body of text. They present conditional expectation through Hilbert space approach and Radon-Nikodym theorem is proved at the end of the book using martingales. This is an indoctrinated way of showing how martingales are applied in other field of mathematics. Each chapter starts with pedagogical explanation of concept and summary of results. This helps reader grasp concepts and develop intuition. The topics, examples and exercises are carefully chosen and well organized. I found several but minor typos and discrepancy in the notation during the last five chapters. Yes, elegant proof is given for each theorem on martingales but rephrasing them may help make it clear where in the proof previous results are used and applied. Also, it would be a great idea to include introductory texts on stochastic calculus in the reference for the beginning students. Despite these minor suggestions, I recommend the book with enthusiasm. After reading this book, one can take their way immediately to stochastic calculus: Brownian motion and Ito calculus and their applications.

All backgound needed for Ito calculus is here!.......2000-10-04

This is an excellent and timely textbook on probability and martingale theory. There is an increasing need of thorough but concise treatise of probability theory for researchers and graduate students in Engineering, Economics, Statistics and Mathematical Biology. Very few textbook fill this need. Jacod and Protter succeeded in bringing together essential concepts and theorems in probability/martingale theory in a clear and lucid style and the book is completely self-contained: all necessary machinery from measure theory are explained and proved while providing a flavor of probabilistic way of thinking. Unlike Williams' "Probability with Martingales", all mathematical details are covered in the body of text. They present conditional expectation through Hilbert space approach and Radon-Nikodym theorem is proved at the end of the book using martingales. This is an indoctrinated way of showing how martingales are applied in other field of mathematics. Each chapter starts with pedagogical explanation of concept and summary of results. This helps reader grasp concepts and develop intuition. The topics, examples and exercises are carefully chosen and well organized. I found several but minor typos and discrepancy in the notation during the last five chapters. Yes, elegant proof is given for each theorem on martingales but rephrasing them may help make it clear where in the proof previously results are used and applied. Also, it would be a great idea to include introductory texts on stochastic calculus for the beginning students. Despite these minor suggestions, I recommend the book with enthusiasm. After reading this book, one can take their way immediately to stochastic calculus: Brownian motion and Ito calculus.

The Elements of Integration and Lebesgue Measure

Average customer rating:

A good introduction: concise and clear.
Good Integration and Measure Into (A Bit Expensive Though)
IF YOU WANT TO UNDERSTAND MEASURE THEORY...
Excellent as an itroduction and as a reference
A great place to begin

The Elements of Integration and Lebesgue Measure
Robert G. Bartle
Manufacturer: Wiley-Interscience
ProductGroup: Book
Binding: Paperback

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ASIN: 0471042226

Book Description

The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: T. W. Anderson The Statistical Analysis of Time Series T. S. Arthanari & Yadolah Dodge Mathematical Programming in Statistics Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences Robert G. Bartle The Elements of Integration and Lebesgue Measure George E. P. Box & George C. Tiao Bayesian Inference in Statistical Analysis R. W. Carter Simple Groups of Lie Type William G. Cochran & Gertrude M. Cox Experimental Designs, Second Edition Richard Courant Differential and Integral Calculus, Volume I Richard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II D. R. Cox Planning of Experiments Harold M. S. Coxeter Introduction to Modern Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II Bruno de Finetti Theory of Probability, Volume 1 Bruno de Finetti Theory of Probability, Volume 2 W. Edwards Deming Sample Design in Business Research Amos de Shalit & Herman Feshbach Theoretical Nuclear Physics, Volume 1 —Nuclear Structure J. L. Doob Stochastic Processes Nelson Dunford & Jacob T. Schwartz Linear Operators, Part One, General Theory Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Two, Spectral Theory—Self Adjoint Operators in Hilbert Space Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Three, Spectral Operators Herman Feshbach Theoretical Nuclear Physics: Nuclear Reactions Bernard Friedman Lectures on Applications-Oriented Mathematics Phillip Griffiths & Joseph Harris Principles of Algebraic Geometry Gerald J. Hahn & Samuel S. Shapiro Statistical Models in Engineering Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume I—Methods and Applications Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume II—Theory Peter Henrici Applied and Computational Complex Analysis, Volume 1—Power Series—Integration—Conformal Mapping—Location of Zeros Peter Henrici Applied and Computational Complex Analysis, Volume 2—Special Functions—Integral Transforms—Asymptotics—Continued Fractions Peter Henrici Applied and Computational Complex Analysis, Volume 3—Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions Peter Hilton & Yel-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin O. Kreyszig Introductory Functional Analysis with Applications William H. Louisell Quantum Statistical Properties of Radiation Ali Hasan Nayfeh Introduction to Perturbation Techniques Emanuel Parzen Modern Probability Theory and Its Applications P. M. Prenter Splines and Variational Methods Walter Rudin Fourier Analysis on Groups C. L. Siegel Topics in Complex Function Theory, Volume I—Elliptic Functions and Uniformization Theory C. L. Siegel Topics in Complex Function Theory, Volume II—Automorphic and Abelian Integrals C. L. Siegel Topics in Complex Function Theory, Volume III—Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry J. J. Stoker Water Waves: The Mathematical Theory with Applications J. J. Stoker Nonlinear Vibrations in Mechanical and Electrical Systems

Customer Reviews:

A good introduction: concise and clear........2007-01-27

The book is concise and easy to follow. The author rarely gives lengthy explanations and analogies, but spends the bulk of the book stating solid facts and proofs. I also like the organization of the book. All definitions and theorems are explicitly stated and indexed, not scattered in paragraphs in the body of the text.

The book misses subjects such as complex measures (they are briefly mentioned), the fundamental theorem of calculus under Lebesgue settings, and probability measures, but its ok since the book is an introduction to the subject. A more comprehensive (and harder to read) book is "Real & Complex Analysis" by Walter Rudin. If you are interested in probability, consider Ptrick Billingsley's book "Probability and Measure".

Good Integration and Measure Into (A Bit Expensive Though).......2005-01-15

The exposition of integration in this book is the clearest I have read. I also found the chapter on modes of convergence, where it laid out the relationship between things such as L^P-convergence and convergence in measure, to be extremely useful. The second half, where it covers topics like Lebesgue measure, repeats some of the same information from the first part which is a bit iritating if you are reading straight throught, but contains a lot of good information. The book is also quite small making it easy to take with you as a quick reference.

Let me warn you though that this is an introduction to integration and measure _not_ an introduction to real analysis. It does not cover important topics like L^P-approximation, differentiation, etc. For a complete treatment of real analysis, I recommend the books "Lebesgue Integration on Euclidean Space" by Frank Jones and the slightly more abstract "Real and Functional Analysis" by Serge Lange.

IF YOU WANT TO UNDERSTAND MEASURE THEORY..........2001-06-04

IF YOU WANT TO UNDERSTAND MEASURE THEORY READ THIS BOOK, MAYBE THE ONLY PROBLEM IS THE LACK OF EXAMPLES BUT THE WAY THAT THE THEORY IS PRESENTED MAKE IT YOUR FIRST CHOICE WHEN YOU TRY TO LEARN MEASURE THEORY.

Excellent as an itroduction and as a reference.......2000-03-31

When I took my first one-semester course on measure and Lebesgue integration my teacher chose Bartle's "The Elements of Integration" as text. After reading many other books on the subject now I'm sure he made a wise decision.

Assuming almost no strong mathematical background, Bartle is able to build up the basic Lebesgue integral theory introducing the fundamental abstract concepts (sigma-algebra, measurable function, measure space, "almost everywhere", step function, etc.) in such an easy way that the student is not only able to handle them but to UNDERSTAND them.

From the first part of the book I appreciate specially chapters 6, 7, and 10, on L_p spaces, modes of convergence, and product measures, respectively. These chapters contain the most used results of the basic theory, and they are stated exactly in the way one needs them, making the book very useful for future reference.

I like the second part very much also, because it stresses the importance of measure theory by itself and not only as a requisite for integration theory. If you are interested in fractal geometry or geometric measure theory you will find chapters 11 to 17 very helpful.

Since I own this book it has never been lazy in my bookshelf.

A great place to begin.......2000-02-04

Measure and Integration is a daunting subject for mathematical neophytes. Bartle's little volume is the right place to start. I first learned measure theory from it 20 years ago and went on to study functional analysis and stochastic approximation.

Book Description

Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately. Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset. In addition, the text provides students a through treatment of Fourier analysis, while holistically preparing students to become "workers" in real analysis.

Customer Reviews:

great!.......2006-03-30

This is a terrific text for a first course in graduate-level real analysis, and is suitable for self-study. It develops Lebesgue integration theory slowly, in a very clear manner. In addition, the latter part of the book covers the basics of Fourier Analysis and important topics in differentiation. I frequently refer to this book, as the results are easy to find.

Rigor not Rigor Mortis.......2006-02-25

One of the problems with modern mathematics is its obsession with rigor which has been attended, over the last few decades, by a mushrooming of symbols and jargon. Much of it is not clearly related to the ideas they serve to label, as evidenced by such terms as the topological use of "filter" whose etymology is obscure (ascribed by some to H. Cartan). Moreover, the particular subject of Lebesgue integration and its generalizations is made even more confusing by a wide variety of approaches depending on an author's penchants--many of whom are enamored with a purely axiomatic approach and who make little or no appeal to intuition or--God forbid!--pictures. The author of the present work is obviously someone who has actually taught mathematics and taught it lovingly. This book is an excellent read with lots of interesting topics well explained from a student's point of view. There seems to be a nice ramping from the truly elementary to the sophisticated, which means the book will interest experienced mathematicians, scientists and engineers. There are lots of "doable" problems that the reader can solve along the way. For the experienced mathematician these little problems help alot as a refresher (Oh!, now I remember, that's how you do it.). I like the emphasis on Euclidean space. Somehow, I always feel more comfortable there! It gives me things I can actually construct and doodle on paper. And, it allows the author to use a few figures in a meaningful way. Which is another of the book's strong points and if I could recommend a future improvement, it would be to bring on more of those pictures! Tristram Needham has done a nice job along these lines with his book "Visual Complex Analysis." (I ordered several copies as Christmas gifts--just kidding!). Anyone who has taught mathematics and genuinely wished to be understood by his students has, at various times, drawn them pictures. Inside the cover sheets are lists of integration formulae, a fourier transform table, and a table of "assorted facts" on things like the Gamma function; which show that this is not only a book on Lebesgue integration but a calculus book with the Lebesgue integral occupying center stage. Everyone who has been enamored by the notion of the integral--as I was as a freshman calculus student and have been ever since--will want to have this book on their shelf.

an excellent introductory text.......2003-09-24

As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.

The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.

For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.

High Praise for Jones.......2000-08-22

"Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.

The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.

Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.

While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!

As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.

treasure trove of mathematical technique.......2000-04-01

This book is a treasure trove of mathematical technique. It covers topics that are relevant to many broad areas of real and functional analysis including signal processing and approximation theory. The author takes the time not only to prove the results, but also to construct the proofs so that the technique is made explicit to the reader. The author also motivates definitions by breaking them into the successively more complicated pieces so as to build intuition in the reader.

I especially recommend this book to anyone who lacks formal training in mathematics or wishes to develop mathematical technique in the areas of real and functional analysis.

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